Random potentials for pinning models with \nabla and \Delta interactions

TL;DR

This paper studies two polymer models with Gaussian potentials and random environments, analyzing how the critical points differ between annealed and quenched cases despite model differences.

Contribution

It introduces two new biopolymer models with gradient and Laplacian interactions in random environments and compares their critical points to classical pinning models.

Findings

The gap between annealed and quenched critical points remains consistent across models.

The models exhibit similar phase transition behavior to classical pinning models.

Random charges influence the polymer's interaction with the origin.

Abstract

We consider two models for biopolymers, the $\nabla$ interaction and the $\Delta$ one, both with the Gaussian potential in the random environment. A random field $\varphi:{0,1,...,N}\rightarrow \Bbb{R}^d$ represents the position of the polymer path. The law of the field is given by $\exp(-\sum_i\frac{|\nabla\varphi_i|^2}{2})$ where $\nabla$ is the discrete gradient, and by $\exp(-\sum_i\frac{|\Delta\varphi_i|^2}{2})$ where $\Delta$ is the discrete Laplacian. For every Gaussian potential $\frac{|\cdot|^2}{2}$, a random charge is added as a factor: $(1+\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\Bbb{P}(\omega_i=\pm 1)=1/2$ or $\exp(\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\omega_i$ obeys a normal distribution. The interaction with the origin in the random field space is considered. Each time the field touches the origin, a reward $\epsilon\geq 0$ is given. Although these models are quite different from the pinning models studied in Giacomin (2007), the result about the gap between the annealed critical point and the quenched critical point stays the same.